Finite Group Databasebeta

← C3×#86273 #11901760

Subgroups — C3×#86273 #11901760

Order 1152.

2191 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#533C211yesmaximal, characteristic, complemented
2C3×#5733C214yesmaximal, complemented
3C3×#5698C212yesmaximal, complemented
4C3×#5697C212yesmaximal, complemented
5C3×#5645C214yesmaximal, complemented
6C3×#5644C214yesmaximal, complemented
7C3×#5580C218yesmaximal, complemented
8C3×#5524C212yesmaximal, complemented
9C3×#5522C212yesmaximal, complemented
10C3×#5477C211yesmaximal, characteristic, complemented
11C2×C3×#872C211yesmaximal, characteristic, complemented
12C3×#3455—31nomaximal
13#86273C311yesmaximal, characteristic, direct product factor
14C2×C3×#235C2212yes—
15C22×C3×C4×D3C2211yescharacteristic
16C32×C4⋊Q8C2212yescomplemented
17C32×#125C2212yescomplemented
18C32×C22⋊Q8C2214yes—
19C32×C4×Q8C2214yescomplemented
20C32×#117C2212yes—
21C2×C32×C42C2211yescharacteristic
22C3×#947C2218yes—
23C3×#946C2218yescomplemented
24C3×#930C2214yes—
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×C4×Dic3C2211yescharacteristic
28C3×#898C2218yes—
29C3×#898C2218yescomplemented
30C3×#897C2214yes—
31C3×#895C2214yes—
32C3×#895C2218yes—
33C3×#894C2218yes—
34C3×#894C2218yescomplemented
35C3×#893C2214yescomplemented
36C3×#890C2218yescomplemented
37C3×#887C2214yescomplemented
38C3×#886C2214yes—
39C3×#882C2214yes—
40C3×#67⋊C4C2218yes—
41C3×#875C2212yescomplemented
42C3×#874C2211yescharacteristic, complemented
43C3×C4×D12C2218yes—
44C3×C4×D12C2214yescomplemented
45C3×#872C2214yes—
46C3×#872C2212yes—
47C3×#872C2212yes—
48C3×#872C2214yescomplemented
49C3×C42×D3C2214yescomplemented
50C3×#869C2211yescharacteristic, complemented
51C3×#542—34no—
52C3×#541—32no—
53C3×#540—32no—
54C3×#539—34no—
55C3×#538—34no—
56C3×#533—21no—
57C3×#533D311yescharacteristic, complemented
58C3×#533C2×C311yescharacteristic, complemented
59C3×#532—31no—
60C3×#529—38no—
61C3×#525—32no—
62C3×C4×#41—32no—
63C2×C3×#117—31no—
64#5733C2×C314yescomplemented
65#5698C2×C312yescomplemented
66#5697C2×C312yescomplemented
67#5645C2×C314yescomplemented
68#5644C2×C314yescomplemented
69#5580C2×C318yescomplemented
70#5524C2×C312yescomplemented
71#5522C2×C312yescomplemented
72#5477C2×C311yescharacteristic, complemented
73C2×#872C2×C311yescharacteristic, complemented
74C23×C3×D3C2311yescharacteristic
75C2×C32×Q8C2314yes—
76C22×C32×C4C2312yes—
77C22×C32×C4C2311yescharacteristic
78C2×C3×#67C2314yes—
79C22×C3×Dic3C2311yescharacteristic
80C22×C3×Dic3C2312yes—
81C3×#235—28no—
82C2×C3×D12C2312yes—
83C2×C3×D12C2318yes—
84C2×C3×D12C2314yes—
85C2×C3×C4×D3—24no—
86C2×C3×C4×D3C2312yes—
87C2×C3×C4×D3C2312yes—
88C2×C3×C4×D3C2314yes—
89C2×C3×C4×D3C2318yes—
90C2×C3×Dic6C2312yes—
91C32×C4⋊C4C2314yes—
92C32×C4⋊C4C2318yes—
93C32×C4⋊C4C2314yes—
94C32×C22⋊C4C2314yes—
95C32×C42C2312yes—
96C32×C42C2311yescharacteristic
97C32×C42C2311yescharacteristic
98C32×C42C2314yes—
99C3×(C2×D3)⋊C4C2318yes—
100C3×(C2×D3)⋊C4C2318yes—
101C3×(C2×D3)⋊C4C2318yes—
102C3×(C2×D3)⋊C4C2314yes—
103C3×(C2×D3)⋊C4C2318yes—
104C3×(C2×D3)⋊C4C2314yes—
105C3×C4⋊Dic3C2318yes—
106C3×C4⋊Dic3C2314yes—
107C3×Dic3⋊C4C2318yes—
108C3×Dic3⋊C4C2314yes—
109C3×Dic3⋊C4C2314yes—
110C3×C4×Dic3C2318yes—
111C3×C4×Dic3C2312yes—
112C3×C4×Dic3C2312yes—
113#3455—31noSylow
114C2×C3×#41—32no—
115C23×C3×C4—31no—
116C2×#235C22×C312yes—
117C22×C4×D3C22×C311yescharacteristic
118C3×C4⋊Q8—22no—
119C3×C4⋊Q8—31no—
120C3×C4⋊Q8C2×D312yescomplemented
121C3×C4⋊Q8C22×C312yescomplemented
122C3×#127—31no—
123C3×#126—38no—
124C3×#126—38no—
125C3×#125—22no—
126C3×#125—38no—
127C3×#125C2×D312yescomplemented
128C3×#125C22×C312yescomplemented
129C3×#124—38no—
130C3×#124—32no—
131C3×#123—38no—
132C3×#123—38no—
133C3×#123—34no—
134C3×C22⋊Q8—24no—
135C3×C22⋊Q8—38no—
136C3×C22⋊Q8—34no—
137C3×C22⋊Q8C2×D314yes—
138C3×C22⋊Q8C22×C314yes—
139C3×#121—34no—
140C3×#121—34no—
141C3×#121—38no—
142C3×#120—34no—
143C3×C4×Q8—24no—
144C3×C4×Q8—34no—
145C3×C4×Q8C2×D314yescomplemented
146C3×C4×Q8C22×C314yescomplemented
147C3×C4×D4—34no—
148C3×C4×D4—38no—
149C3×C4×D4—38no—
150C3×C4×D4—34no—
151C3×#117—22no—
152C3×#117—34no—
153C3×#117—32no—
154C3×#117—32no—
155C3×#117—34no—
156C3×#117C2×D312yes—
157C3×#117C22×C312yes—
158C2×C3×C4⋊C4—32no—
159C2×C3×C22⋊C4—32no—
160C2×C3×C42—21no—
161C2×C3×C42—31no—
162C2×C3×C42—34no—
163C2×C3×C42C2×D311yescharacteristic
164C2×C3×C42C22×C311yescharacteristic
165#947C22×C318yes—
166#946C22×C318yescomplemented
167#930C22×C314yes—
168C2×(C2×D3)⋊C4C22×C312yes—
169C2×Dic3⋊C4C22×C312yes—
170C2×C4×Dic3C22×C311yescharacteristic
171#898C22×C318yes—
172#898C22×C318yescomplemented
173#897C22×C314yes—
174#895C22×C318yes—
175#895C22×C314yes—
176#894C22×C318yes—
177#894C22×C318yescomplemented
178#893C22×C314yescomplemented
179#890C22×C318yescomplemented
180#887C22×C314yescomplemented
181#886C22×C314yes—
182#882C22×C314yes—
183#67⋊C4C22×C318yes—
184#875C22×C312yescomplemented
185#874C22×C311yescharacteristic, complemented
186C4×D12C22×C318yes—
187C4×D12C22×C314yescomplemented
188#872C22×C312yes—
189#872C22×C312yes—
190#872C22×C314yes—
191#872C22×C314yescomplemented
192C42×D3C22×C314yescomplemented
193#869C22×C311yescharacteristic, complemented
194C23×C32C2411yescharacteristic
195C22×C3×D3—22no—
196C22×C3×D3—42no—
197C22×C3×D3C2412yes—
198C22×C3×D3C2414yes—
199C32×Q8—44no—
200C2×C32×C4—42no—
201C2×C32×C4#4114yes—
202C2×C32×C4C2418yes—
203C2×C32×C4C2412yes—
204C2×C32×C4C2412yes—
205C2×C32×C4C2411yescharacteristic
206C2×C32×C4C2411yescharacteristic
207C3×#67—28no—
208C2×C3×Dic3—22no—
209C2×C3×Dic3—42no—
210C2×C3×Dic3C2414yes—
211C2×C3×Dic3C2414yes—
212C2×C3×Dic3C2412yes—
213C3×D12—24no—
214C3×D12—48no—
215C3×D12—44no—
216C3×C4×D3—24no—
217C3×C4×D3—24no—
218C3×C4×D3—28no—
219C3×C4×D3—48no—
220C3×Dic6—24no—
221#542—34no—
222#541—32no—
223#540—32no—
224#539—34no—
225#538—34no—
226#533C3×D311yescharacteristic, complemented
227#532—31no—
228#529—38no—
229#525—32no—
230C4×#41—32no—
231C2×#117—31no—
232C24×C3—31no—
233C23×D3C23×C311yescharacteristic
234C3×#41—68no—
235C2×C3×Q8—24no—
236C2×C3×Q8—32no—
237C2×C3×Q8C22×D314yes—
238C2×C3×Q8C23×C314yes—
239C2×C3×D4—34no—
240C2×C3×D4—32no—
241C2×C3×D4—38no—
242C2×C3×D4—34no—
243C22×C3×C4—21no—
244C22×C3×C4—22no—
245C22×C3×C4—31no—
246C22×C3×C4—32no—
247C22×C3×C4—32no—
248C22×C3×C4—34no—
249C22×C3×C4—32no—
250C22×C3×C4—38no—
251C22×C3×C4—64no—
252C22×C3×C4C22×D312yes—
253C22×C3×C4C22×D311yescharacteristic
254C22×C3×C4C23×C312yes—
255C22×C3×C4C23×C311yescharacteristic
256C2×#67C23×C314yes—
257C22×Dic3C23×C311yescharacteristic
258C22×Dic3C23×C312yes—
259#235—28no—
260C2×D12C23×C312yes—
261C2×D12C23×C318yes—
262C2×D12C23×C314yes—
263C2×C4×D3—24no—
264C2×C4×D3C23×C314yes—
265C2×C4×D3C23×C312yes—
266C2×C4×D3C23×C312yes—
267C2×C4×D3C23×C318yes—
268C2×Dic6C23×C312yes—
269C3×C4⋊C4—24no—
270C3×C4⋊C4—28no—
271C3×C4⋊C4—24no—
272C3×C4⋊C4—34no—
273C3×C4⋊C4—34no—
274C3×C4⋊C4—38no—
275C3×C4⋊C4—34no—
276C3×C4⋊C4—38no—
277C3×C4⋊C4C22×D314yes—
278C3×C4⋊C4C22×D318yes—
279C3×C4⋊C4C22×D314yes—
280C3×C4⋊C4C23×C314yes—
281C3×C4⋊C4C23×C318yes—
282C3×C4⋊C4C23×C314yes—
283C3×C22⋊C4—24no—
284C3×C22⋊C4—34no—
285C3×C22⋊C4—34no—
286C3×C22⋊C4—38no—
287C3×C22⋊C4—38no—
288C3×C22⋊C4—38no—
289C3×C22⋊C4—38no—
290C3×C22⋊C4C22×D314yes—
291C3×C22⋊C4C23×C314yes—
292C3×C42—22no—
293C3×C42—21no—
294C3×C42—21no—
295C3×C42—24no—
296C3×C42—32no—
297C3×C42—38no—
298C3×C42—32no—
299C3×C42C22×D311yescharacteristic
300C3×C42C22×D311yescharacteristic
301C3×C42C22×D314yes—
302C3×C42C22×D312yes—
303C3×C42C23×C311yescharacteristic
304C3×C42C23×C314yes—
305C3×C42C23×C312yes—
306C3×C42C23×C311yescharacteristic
307(C2×D3)⋊C4C23×C314yes—
308(C2×D3)⋊C4C23×C314yes—
309(C2×D3)⋊C4C23×C318yes—
310(C2×D3)⋊C4C23×C318yes—
311(C2×D3)⋊C4C23×C318yes—
312(C2×D3)⋊C4C23×C318yes—
313C4⋊Dic3C23×C314yes—
314C4⋊Dic3C23×C318yes—
315Dic3⋊C4C23×C314yes—
316Dic3⋊C4C23×C314yes—
317Dic3⋊C4C23×C318yes—
318C4×Dic3C23×C312yes—
319C4×Dic3C23×C318yes—
320C4×Dic3C23×C312yes—
321C22×C32—22no—
322C22×C32C2×#4112yes—
323C22×C32C2511yescharacteristic
324C2×C3×D3—22no—
325C2×C3×D3—24no—
326C2×C3×D3—28no—
327C2×C3×D3—24no—
328C2×C3×D3—44no—
329C32×C4—22no—
330C32×C4—28no—
331C32×C4—22no—
332C32×C4C2×#4112yes—
333C32×C4C2×#4112yes—
334C3×Dic3—24no—
335C3×Dic3—22no—
336C3×Dic3—24no—
337C2×#41—32no—
338C23×C4—31no—
339C4⋊Q8—31no—
340C4⋊Q8C2×C3×D312yescomplemented
341#127—31no—
342#126—38no—
343#126—38no—
344#125—38no—
345#125C2×C3×D312yescomplemented
346#124—38no—
347#124—32no—
348#123—38no—
349#123—38no—
350#123—34no—
351C22⋊Q8—38no—
352C22⋊Q8—34no—
353C22⋊Q8C2×C3×D314yes—
354#121—34no—
355#121—34no—
356#121—38no—
357#120—34no—
358C4×Q8—34no—
359C4×Q8C2×C3×D314yescomplemented
360C4×D4—34no—
361C4×D4—38no—
362C4×D4—38no—
363C4×D4—34no—
364#117—34no—
365#117—32no—
366#117—32no—
367#117—34no—
368#117C2×C3×D312yes—
369C2×C4⋊C4—32no—
370C2×C22⋊C4—32no—
371C2×C42—31no—
372C2×C42—34no—
373C2×C42C2×C3×D311yescharacteristic
374C23×C3—21no—
375C23×C3—32no—
376C23×C3—34no—
377C23×C3—62no—
378C23×C3—122no—
379C23×C3C23×D311yescharacteristic
380C23×C3C24×C311yescharacteristic
381C22×D3—22no—
382C22×D3—42no—
383C22×D3C24×C312yes—
384C22×D3C24×C314yes—
385C3×Q8—44no—
386C3×Q8—44no—
387C3×Q8—48no—
388C3×Q8—64no—
389C3×D4—68no—
390C3×D4—64no—
391C3×D4—124no—
392C3×D4—128no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—28no—
398C2×C3×C4—24no—
399C2×C3×C4—34no—
400C2×C3×C4—34no—
401C2×C3×C4—32no—
402C2×C3×C4—42no—
403C2×C3×C4—42no—
404C2×C3×C4—44no—
405C2×C3×C4—62no—
406C2×C3×C4—64no—
407C2×C3×C4—64no—
408C2×C3×C4—68no—
409C2×C3×C4—128no—
410C2×C3×C4—122no—
411C2×C3×C4#23914yes—
412C2×C3×C4C3×#4114yes—
413C2×C3×C4C23×D318yes—
414C2×C3×C4C23×D312yes—
415C2×C3×C4C23×D312yes—
416C2×C3×C4C23×D311yescharacteristic
417C2×C3×C4C23×D311yescharacteristic
418C2×C3×C4C24×C318yes—
419C2×C3×C4C24×C312yes—
420C2×C3×C4C24×C312yes—
421C2×C3×C4C24×C311yescharacteristic
422C2×C3×C4C24×C311yescharacteristic
423#67—28no—
424C2×Dic3—22no—
425C2×Dic3—42no—
426C2×Dic3C24×C314yes—
427C2×Dic3C24×C314yes—
428C2×Dic3C24×C312yes—
429D12—24no—
430D12—48no—
431D12—44no—
432C4×D3—24no—
433C4×D3—24no—
434C4×D3—28no—
435C4×D3—48no—
436Dic6—24no—
437C2×C32—22no—
438C2×C32C22×#4111yescharacteristic
439C2×C32#58512yes—
440C3×D3—44no—
441C3×D3—42no—
442C24—31no—
443#41—68no—
444C2×Q8—32no—
445C2×Q8C22×C3×D314yes—
446C2×D4—34no—
447C2×D4—32no—
448C2×D4—38no—
449C2×D4—34no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—34no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—38no—
456C22×C4—64no—
457C22×C4C22×C3×D312yes—
458C22×C4C22×C3×D311yescharacteristic
459C4⋊C4—34no—
460C4⋊C4—34no—
461C4⋊C4—38no—
462C4⋊C4—34no—
463C4⋊C4—38no—
464C4⋊C4C22×C3×D314yes—
465C4⋊C4C22×C3×D318yes—
466C4⋊C4C22×C3×D314yes—
467C22⋊C4—34no—
468C22⋊C4—34no—
469C22⋊C4—38no—
470C22⋊C4—38no—
471C22⋊C4—38no—
472C22⋊C4—38no—
473C22⋊C4C22×C3×D314yes—
474C42—32no—
475C42—32no—
476C42—38no—
477C42C22×C3×D314yes—
478C42C22×C3×D312yes—
479C42C22×C3×D311yescharacteristic
480C42C22×C3×D311yescharacteristic
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—21no—
485C22×C3—42no—
486C22×C3—62no—
487C22×C3—64no—
488C22×C3—68no—
489C22×C3—64no—
490C22×C3—124no—
491C22×C3C2×#23912yes—
492C22×C3C2×C3×#4112yes—
493C22×C3C24×D311yescharacteristic
494C22×C3C25×C311yescharacteristic
495C2×D3—22no—
496C2×D3—24no—
497C2×D3—28no—
498C2×D3—24no—
499C2×D3—44no—
500C3×C4—24no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—28no—
504C3×C4—28no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—48no—
511C3×C4—62no—
512C3×C4—64no—
513C3×C4—64no—
514C3×C4C2×#23912yes—
515C3×C4C2×#23912yes—
516C3×C4C2×C3×#4112yes—
517C3×C4C2×C3×#4112yes—
518Dic3—24no—
519Dic3—22no—
520Dic3—24no—
521C32#345511yescharacteristic, complemented, Sylow
522C23—32no—
523C23—34no—
524C23—62no—
525C23—122no—
526C23C23×C3×D311yescharacteristic
527Q8—44no—
528Q8—64no—
529D4—68no—
530D4—64no—
531D4—124no—
532D4—128no—
533C2×C4—34no—
534C2×C4—34no—
535C2×C4—32no—
536C2×C4—42no—
537C2×C4—62no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—68no—
541C2×C4—128no—
542C2×C4—122no—
543C2×C4C3×#23914yes—
544C2×C4C23×C3×D318yes—
545C2×C4C23×C3×D312yes—
546C2×C4C23×C3×D312yes—
547C2×C4C23×C3×D311yescharacteristic
548C2×C4C23×C3×D311yescharacteristic
549C2×C3—22no—
550C2×C3—22no—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—122no—
555C2×C3—124no—
556C2×C3C22×#23911yescharacteristic
557C2×C3#596412yes—
558C2×C3C22×C3×#4111yescharacteristic
559C2×C3C3×#58512yes—
560D3—42no—
561D3—44no—
562C22—22no—
563C22—62no—
564C22—64no—
565C22—68no—
566C22—64no—
567C22—124no—
568C22C2×C3×#23912yes—
569C22C24×C3×D311yescharacteristic
570C4—22no—
571C4—22no—
572C4—28no—
573C4—62no—
574C4—64no—
575C4—64no—
576C4C2×C3×#23912yes—
577C4C2×C3×#23912yes—
578C3—21no—
579C3#8627311yescharacteristic, direct product factor
580C3C3×#345511yescharacteristic, complemented
581C2—22no—
582C2—122no—
583C2—124no—
584C2C22×C3×#23911yescharacteristic
585C2C3×#596412yes—
subgroupG/H
centerC2×C3×C4C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#533C2
socleC22×C32C25
hypercenterC3×#533D3
hypocenterC3C3×#3455
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×#533

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C32×#533C2semidirect
2. C3×#5733C2semidirect
3. C3×#5698C2semidirect
4. C3×#5697C2semidirect
5. C3×#5645C2semidirect
6. C3×#5644C2semidirect
7. C3×#5580C2semidirect
8. C3×#5524C2semidirect
9. C3×#5522C2semidirect
10. C3×#5477C2semidirect
11. C2×C3×#872C2semidirect
13. #86273C3direct (N × H)
16. C32×C4⋊Q8C22semidirect
17. C32×#125C22semidirect
19. C32×C4×Q8C22semidirect
23. C3×#946C22semidirect
29. C3×#898C22semidirect
34. C3×#894C22semidirect
35. C3×#893C22semidirect
36. C3×#890C22semidirect
37. C3×#887C22semidirect
41. C3×#875C22semidirect
42. C3×#874C22semidirect
44. C3×C4×D12C22semidirect
48. C3×#872C22semidirect
49. C3×C42×D3C22semidirect
50. C3×#869C22semidirect
57. C3×#533D3semidirect
58. C3×#533C2×C3semidirect
64. #5733C2×C3semidirect
65. #5698C2×C3semidirect
66. #5697C2×C3semidirect
67. #5645C2×C3semidirect
68. #5644C2×C3semidirect
69. #5580C2×C3semidirect
70. #5524C2×C3semidirect
71. #5522C2×C3semidirect
72. #5477C2×C3semidirect
73. C2×#872C2×C3semidirect
120. C3×C4⋊Q8C2×D3semidirect
121. C3×C4⋊Q8C22×C3semidirect
127. C3×#125C2×D3semidirect
128. C3×#125C22×C3semidirect
145. C3×C4×Q8C2×D3semidirect
146. C3×C4×Q8C22×C3semidirect
166. #946C22×C3semidirect
172. #898C22×C3semidirect
177. #894C22×C3semidirect
178. #893C22×C3semidirect
179. #890C22×C3semidirect
180. #887C22×C3semidirect
184. #875C22×C3semidirect
185. #874C22×C3semidirect
187. C4×D12C22×C3semidirect
191. #872C22×C3semidirect
192. C42×D3C22×C3semidirect
193. #869C22×C3semidirect
226. #533C3×D3semidirect
340. C4⋊Q8C2×C3×D3semidirect
345. #125C2×C3×D3semidirect
359. C4×Q8C2×C3×D3semidirect
521. C32#3455semidirect
579. C3#86273direct (N × H)
580. C3C3×#3455semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#5332
C3×#57332
C3×#56982
C3×#56972
C3×#56452
C3×#56442
C3×#55802
C3×#55242
C3×#55222
C3×#54772
C2×C3×#8722
C3×#34553
#862733